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A tubular segmented-in-series solid oxide fuel cell with metallic interconnect films: A performance study through mathematical simulations
Current Applied Physics 13 (2013) 1906e1913
Contents lists available at ScienceDirect
Current Applied Physics journal homepage: www.elsevier.com/locate/cap
A tubular segmented-in-series solid oxide fuel cell with metallic interconnect films: A performance study through mathematical simulations Bu-Won Son a, b,1, Seok-Joo Park a, Seung-Bok Lee a, Tak-Hyoung Lim a, Rak-Hyun Song a, Jong-Won Lee a, b, * a b
New and Renewable Energy Research Division, Korea Institute of Energy Research, 152 Gajeong-ro, Yuseong-gu, Daejeon 305-343, Republic of Korea Department of Advanced Energy Technology, University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 305-350, Republic of Korea
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 January 2013 Received in revised form 11 July 2013 Accepted 31 July 2013 Available online 9 August 2013
In a segmented-in-series solid oxide fuel cell (SIS-SOFC), an interconnect (IC) provides electrical contact and sealing between the anode of one cell and the cathode of the next. A metallic silver-glass composite (SGC) is considered a promising alternative to ceramic IC materials in SIS-SOFCs. In this work, a simulation study is performed on a tubular SIS-SOFC to assess the effectiveness of the SGC-IC design and to predict the SOFC performance characteristics for various IC geometries and conductivities. The developed model provides detailed information on cell behavior, such as the internal resistance, the potential/ current distribution, and the local gas species concentration. The results demonstrate that the SGC material greatly reduces a potential drop across the IC film. Thus, it provides the following substantial advantages over conventional ceramic IC materials: (i) increased power density and (ii) a larger degree of flexibility in the cell design. Moreover, the validation test, i.e., comparison of the simulated results with the experimental data, indicates that the model could serve as a valuable tool for design optimization to achieve the required SOFC performance. Ó 2013 Elsevier B.V. All rights reserved.
Keywords: Solid oxide fuel cell Segmented-in-series Mathematical model Interconnect Silver-glass composite
1. Introduction Solid oxide fuel cells (SOFCs) have received increasing attention in recent years because they have higher thermodynamic efficiency for energy conversion than low-temperature fuel cells (e.g., polymer electrolyte membrane fuel cells). In addition, SOFCs can be operated with a wide range of fuels, including natural gas, liquid hydrocarbons, coals, and bio-derived fuels [1,2]. A segmented-inseries (SIS) SOFC is an advanced design consisting of segmented unit cells connected in both electrical and gas flow series [3]. The unit cells are usually fabricated as a thin banded structure on a porous inert support tube, either tubular or flat-tubular, and they are electrically connected by interconnect (IC) layers [4,5]. The most attractive feature of the SIS-SOFC design is that the series
* Corresponding author. New and Renewable Energy Research Division, Korea Institute of Energy Research, 152 Gajeong-ro, Yuseong-gu, Daejeon, 305-343, Republic of Korea. Tel.: þ82 42 860 3025; fax: þ82 42 860 3297. E-mail address: [email protected] (J.-W. Lee). 1 Present address: Corporate R&D, Future Technology Center, LG Chem, 104-1, Moonji-dong, Yuseong-gu, Daejeon 305-738, Republic of Korea. 1567-1739/$ e see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cap.2013.07.025
connection of multiple cells on a single support tube builds up the total output voltage, leading to improved stack efficiency. In SIS-SOFCs, the IC layer electrically connects segmented unit cells and separates fuel (on the anode side) from oxidant (on the cathode side). The IC layer should be not only impermeable to gases, but also conductive and stable in a dual atmosphere (a reducing atmosphere on the anode side and an oxidizing atmosphere on the cathode side). Most IC films in SIS-SOFCs to date have been made of ceramic materials, such as doped LaCrO3 or SrTiO3 perovskites. Ca or Sr-doped LaCrO3 perovskites show acceptable conductivity and stability in the fuel cell environment [6,7]; however, it is quite difficult to fabricate a dense IC film on a porous support due to the poor sinterability of chromites. La-doped SrTiO3 perovskites have been adopted as an alternative IC material to doped chromites, as in the tubular SIS-SOFCs developed by Mitsubishi Heavy Industries Ltd. [5]. However, doped titanates are known to exhibit quite low electrical conductivity in a dual atmosphere, resulting in an increased internal resistance [8]. To address these problems, we recently developed a new tubular SIS-SOFC design, in which segmented unit cells are electrically connected by metallic IC films [9]. The IC films are fabricated
B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
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Fig. 1. Schematic diagrams of tubular SIS-SOFCs with (a) ceramic IC films and (b) SGC-IC films used for the simulation analysis. The drawings are not to scale.
using a silver-glass composite (SGC) material that is known as a conductive sealant in SOFCs [10,11]. Our previous study [11] showed that the electrical conductivity of the SGC material varies between 8.3 103 and 3.5 104 S m1 at 1023 K, depending on the glass content in the composite, while providing gas-tight sealing. Furthermore, the feasibility of the SGC-IC film was demonstrated on a tubular SIS-SOFC having 10 segmented cells. A power density as high as 3.3 103 W m2 (based on the active area) was achieved at 1023 K, and there was no performance degradation over 1100 h of continuous operation at 923 K [9]. No comprehensive optimization study was carried out, but it is expected that further optimizations of the cell and IC designs will lead to improved cell performance. The information provided by the previous experimental work [9] was limited to the fabrication procedure and the overall performance of the SIS-SOFC with metallic SGC-IC films. A deeper understanding of physical and electrochemical phenomena relevant to cell performance is still lacking. Modeling studies would be helpful in understanding the significance of the new IC design features and in determining the optimal IC geometries and the performance characteristics. With the objective of providing more information on the new IC design and the cell performance, this paper reports a modeling study of the tubular SIS-SOFC with metallic SGC-IC films. Firstly, a mathematical model for a tubular SIS-SOFC was developed that considers the electrochemical reaction, charge transport, and gas-phase mass transport. Secondly, the cell behavior was simulated for conventional ceramic IC films and metallic SGC-IC films to assess the effectiveness of the SGC-IC. Thirdly, the simulation was performed to determine the effects of IC geometry and conductivity on the performance, thereby making
predictions concerning optimal geometries and materials properties. Finally, the simulation data were compared with the experimental results for the SIS-SOFC fabricated using SGC-IC films. The distributions of electrical potential and gas species concentration in the SIS-SOFC, which cannot be determined by the experiments,
Table 1 Key parameters used in the simulation model. Parameter Cell geometries Unit cell length, lcell (mm) Anode gap length, lag (mm) Cathode gap length, lcg (mm) Support thickness, ts (mm) Anode thickness, ta (mm) Electrolyte thickness, te (mm) Cathode (1) thickness, tc1 (mm) Cathode (2) thickness, tc2 (mm) Outer radius of the support, rs (mm) Materials and electrochemical properties Anode conductivity, sa (S m1) Electrolyte conductivity, se (S m1) Cathode (1) conductivity, sc1 (S m1) Cathode (2) conductivity, sc2 (S m1) Transfer coefficient, a Anodic pre-exponential coefficient, ga (A m2) Cathodic pre-exponential coefficient, gc (A m2) Anodic activation energy, Eact,a (kJ mol1) Cathodic activation energy, Eact,c (kJ mol1) Tortuosity, s Porosity, 3 Average pore diameter, dp (mm)
Value 8 0.25 0.25 1.8 20 20 15 15 5 1 105 2 5 103 1.5 104 0.5 5.5 1010 7.0 109 120 120 2.5 0.45 1
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were quantitatively conditions.
B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
estimated
under
real
SOFC
operating
2. Cell geometries and simulation model Fig. 1(a) and (b) shows schematic diagrams of the tubular SISSOFCs having the ceramic and SGC-IC films, respectively, used for the simulation analysis. Each unit cell is divided into two regions: (i) the active region where the anode, the electrolyte, and the cathode overlap, and (ii) the inactive region consisting of the anode gap, the interconnect, and the cathode gap. The cathode is assumed to be composed of two layers, designated as cathode (1) and cathode (2). Cathode (1) represents a cathode functional layer in which the electrochemical oxygen reaction takes place, and cathode (2) corresponds to a current collecting layer to facilitate the lateral current flow in the plane of the cathode. Such a bi-layered cathode design has typically been employed in SIS-SOFCs to reduce ohmic losses in the cathode [9,12]. As shown in Fig. 1, the inactive part in the segmented unit cell varies in configuration, depending on whether ceramic material or SGC is used for the IC film. This is mainly due to a restriction on the fabrication process imposed by the sintering temperature of the IC material used: (i) The sintering temperatures of ceramic IC materials, such as doped LaCrO3 and SrTiO3 (1573e1873 K) are generally higher than those of cathode materials (1273e1473 K). For this reason, the ceramic IC film is co-sintered with the support, anode, and electrolyte, followed by the cathode coating and sintering. The ceramic IC film is, therefore, sandwiched between the anode and the cathode, and its entire surface is covered by the cathode, as presented in Fig. 1(a).
(ii) The SGC-IC film, on the other hand, should be coated and sintered after cathode sintering, because the melting point of silver (1235 K) is lower than the sintering temperatures of cathode materials. The SGC-IC film is fabricated so that it is in contact with the cross-sectional area of the cathode, as illustrated in Fig. 1(b). Unless otherwise noted, the anode gap length (lag) and the cathode gap length (lcg) are fixed as 0.25 mm as well, which could be practically achieved using a screen-printing technique. The outer radius (rs) and thickness (ts) of the porous support tube are 5 mm and 1.8 mm, respectively. The fuel is supplied to a channel on the left-hand side of the support tube. It diffuses through the porous support toward the anode, while the oxidant flows over the cathode. During fuel cell operations, the current flows along the anode of the cell, across the electrolyte, and along the cathode. The current then goes across the IC layer to the anode of the next cell. Given the radial symmetry of the tubular design, a 2dimensional model was used. The partial differential equations were described in cylindrical coordinates and solved in 2 dimensions (radial and axial) under the assumption that there are no significant circumferential variations in the boundary conditions. The mathematical model involves (i) the electrochemical reaction at the anode/electrolyte and electrolyte/cathode interfaces, (ii) electronic conduction in the electrodes, the interconnects, and the current collectors, (iii) ionic conduction in the electrolyte, and (iv) gas-phase diffusion in the electrodes, and (v) gas-phase diffusione convection in the channels. The governing and constitutive equations used in the model are summarized in the Appendix. A finite element computational package, COMSOL MULTIPHYSICSÒ, was used to solve the non-linear system of equations. The simulations
Fig. 2. Color maps of the electrical potential distribution in the inactive region of the anode gap and IC film simulated for (a) the ceramic IC film and (b) the SGCeIC film. The current path is also indicated as a streamline. The drawings are not to scale. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
were run at 1023 K using H2 gas with 3 vol.% H2O at the anode and air at the cathode. 3. Fabrication of a tubular SIS-SOFC with SGC-IC films The detailed fabrication process of a tubular SIS-SOFC with SGCIC films can be found elsewhere [9]. Briefly, a tubular support was produced by extrusion of the powder mixture: 3 mol% Y2O3-stabilized ZrO2 (3YSZ), activated carbon (YP-50F), and an organic binder (YB-131D). The extruded support was pre-sintered at 1373 K. An anode layer of NiOe8YSZ was coated onto the presintered support by a dip-coating process and then sintered at 1273 K. An 8YSZ electrolyte layer was formed by a vacuum slurry coating process [13] followed by sintering at 1673 K. A bi-layered cathode composed of (La0.85Sr0.15)0.9MnO3e8YSZ composite and La0.6Sr0.4Co0.2Fe0.8O3 was coated onto the co-sintered tube by a dipcoating process, and then it was sintered at 1423 K. Finally, the IC film was fabricated with the SGC material containing 90 wt.% Ag and 10 wt.% glass. The SGC paste was screen-printed onto the surface of the anode support without the coated electrolyte layer and then sintered at 973 K. 4. Results and discussion To understand how the IC’s design and conductivity affect the cell behavior, firstly, the simulations were performed on the SISSOFCs in which 5 segmented cells were connected in series by
Fig. 3. (a) Electrical potential profiles of the electrodes in the active regions and (b) the values of the current density norm within the electrolyte, calculated for the ceramic and SGC-IC films.
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either the ceramic or SGC-IC films. The key parameters, such as cell geometries and materials/electrochemical parameters, are listed in Table 1. Fig. 2(a) and (b) illustrates the electrical potential distribution (color map) along with the current path (streamline) in the inactive region of the anode gap and IC film, calculated for the ceramic and SGC-IC films, respectively. The values of lactive and lcell were taken as 7.25 and 8.00 mm, respectively, i.e., (lactive/lcell) ¼ 0.9375. As an example, the conductivities (sic) of the ceramic and SGC-IC materials were assumed to be 10 and 3.5 104 S m1, respectively. The simulations were carried out at the unit cell voltage, Vcell ¼ 0.5 V (total out voltage ¼ 2.5 V). The color map of the electrical potential distribution shows that the potential has the lowest value on the anode side where the current enters the unit cell, and it becomes higher on the cathode side where the current exits the cell. A potential difference as large as 0.222 V is observed across the ceramic IC film (Fig. 2(a)). However, the potential distribution across the SGC-IC film becomes more uniform (potential drop ¼ 0.036 V), as shown in Fig. 2(b), even though the thickness of the SGC-IC film (50 mm) is larger than that of the ceramic IC film (20 mm). As seen in Fig. 2, the current goes straight up through the ceramic IC film, whereas the current in the SGC-IC film runs mostly parallel to the xaxis. Fig. 3(a) compares the electrical potential profiles of the electrodes in the active region for the ceramic and SGC-IC films. The (lactive/lcell) value was fixed as 0.9375. The electric potentials of both
Fig. 4. Plots of (a) the maximum power density (Wmax) and (b) the internal cell resistance (Rcell) against the lactive-to-lcell ratio simulated for the SIS-SOFC with the ceramic IC films.
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B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
the cathode and anode decrease with increasing x from 0 to lactive. The potential difference between the cathode and anode at x ¼ lactive is 0.63 V for the SGC-IC film as compared to 0.79 V for the ceramic IC film. This is due to a smaller potential drop across the SGC-IC film as shown in Fig. 2. To gain information about the distribution of the current crossing the electrolyte, the values of the current density norm are plotted against x in Fig. 3(b). The current density flowing across the electrolyte falls with increasing x, and goes through a shallow minimum at x ¼ ca. 2 mm, where the electrode potential difference deviates the least from the open circuit voltage. Then the current density increases with further increasing x, which indicates that most of the current travels through the anode and crosses the electrolyte at larger x to avoid a long current path across the more resistive cathode. The current density for the SGC-IC film remains higher over the whole electrolyte region compared to that for the ceramic IC film. This is ascribed to the fact that, as discussed in Fig. 2, the potential variation across the SGC-IC film is smaller than that across the ceramic IC film. Fig. 4(a) presents the maximum power density (Wmax) of the SIS-SOFC with the ceramic IC films as a function of (lactive/lcell). The unit cell length (lcell) was assumed to be 8 mm. Note that the calculated power was divided by the total area including both the active and inactive regions. The simulations were conducted using typical sic values of doped chromites or titanates (sic ¼ 1e 100 S m1). For sic ¼ 1 S m1, Wmax reaches a maximum value at
Fig. 6. (a) Comparison of the polarization curves of the SIS-SOFC (5 segmented cells) with the SGCeIC films simulated theoretically and measured experimentally, and (b) the color map of the electrical potential distribution simulated under experimental conditions. The drawings (b) are not to scale. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(lactive/lcell) ¼ ca. 0.53 and then decreases with further increasing (lactive/lcell). As sic increases, the (lactive/lcell) value at which Wmax shows a maximum shifts toward a higher value. To acquire a proper understanding about the cell behavior in Fig. 4(a), the internal cell resistance (Rcell) was computed as a function of (lactive/lcell), and the results are presented in Fig. 4(b). Here, Rcell is considered to be the sum of the resistances of the active and inactive parts. As shown in Fig. 4(b), Rcell falls rapidly with increasing (lactive/lcell), which is due to decreased resistance of the active region. When (lactive/lcell) exceeds ca. 0.78, a significant increase in the cell resistance with (lactive/lcell) is observed, particularly, for sic ¼ 1 S m1. This might result from (i) an increased IC resistance (a reduced cross-sectional area of the IC film) and (ii) an increased electrode resistance (an increased electrode length). The occurrence of local maxima on the Wmax vs. (lactive/lcell) curves in Fig. 4(a), therefore, could be understood qualitatively in terms of a trade-off between an increase in the active area and an increase in the resistances of the electrodes and IC film. Fig. 5(a) and (b) shows the plots of Wmax and Rcell vs. (lactive/lcell), respectively, computed with various sic values for the SGC-IC films. Note that Wmax increases continuously with increasing (lactive/lcell) without showing any local maxima, regardless of the sic values used for the simulation. Moreover, Rcell remains almost constant at (lactive/lcell) > ca. 0.16. As a result, the simulation data in Figs. 4 and 5 clearly demonstrate that the SGC-IC film with high electrical conductivity offers two major benefits over the ceramic IC film: (i) increased power density and (ii) a larger degree of flexibility in the cell design (no performance loss for high (lactive/lcell) values).
Table 2 Cell geometries modified in the model for comparison with the experimental data.
Fig. 5. Plots of (a) the maximum power density (Wmax) and (b) the internal cell resistance (Rcell) against the lactive-to-lcell ratio simulated for the SIS-SOFC with the SGCeIC films.
Parameter
Value
Unit cell length, lcell (mm) Active cell length, lactive (mm) Anode gap length, lag (mm) Cathode gap length, lcg (mm)
16 6 1 1
B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
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Fig. 7. Distributions of (a) H2, (b) O2 and (c) H2O species within the fabricated SIS-SOFC with 5 segmented cells under experimental conditions. The drawings are not to scale. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The polarization curve (cell voltage vs. current density) simulated for the SIS-SOFC with the SGC-IC films is compared with the experimental one in Fig. 6(a). The fabricated SIS-SOFC consists of 5 segmented unit cells. Note that the power was divided by the total unit cell area including both the active and inactive regions. The experimental measurements were conducted at 1023 K using the constant flow rate mode with the H2 and air flow rates of 300 and 500 mL min1, respectively. The cell geometries used for the simulation were modified to match those of the fabricated cell, as summarized in Table 2. As shown in Fig. 6, the calculated polarization curve is in good accord with the experimental data, which confirms that the equations and parameters used in the SIS-SOFC model are reasonable. The electrical potential distribution of the fabricated SIS-SOFC was calculated under experimental conditions and is given in Fig. 6(b), in which the potential deceases from the 5th unit cell (right) to the 1st (left). To illustrate the gas species concentrations within the fabricated SIS-SOFC under practical operating conditions, 3-dimensional representations of H2, O2 and H2O concentrations are shown in Fig. 7(a)e(c), respectively. As expected, the highest concentration values are located at the cell inlet, and progressively decreasing H2 and O2 concentrations are observed moving toward the cell outlet. In the channels, the concentration variations of gas species are small due to relatively high mass flow rates, while a steep gradient is observed in the electrode due to consumption (or generation) of gas species and low permeability caused by low effective diffusivity [14].
determine the performance characteristics for various IC geometries and conductivities. The developed model is used to calculate the polarization behavior (potential vs. current curves) as well as the internal resistance, the potential/current distribution, and the local gas species concentration under SOFC operating conditions. The results simulated for ceramic and SGC-IC films indicate that when the SGC material is used for IC films, the potential drop across the IC film is greatly reduced; therefore, the SGC-IC offers advantages over conventional ceramic IC films, namely, increased power density and a larger degree of flexibility in the cell design. The power density is also found to increase with decreasing SGCIC length and with increasing conductivity. The validation test shows that the model precisely predicts the cell behavior; thus, it can be used as a tool for design optimization to improve SOFC performance.
5. Conclusions
The physical model used in this study assumes the following: (i) steady-state conditions; (ii) laminar gas flow in the channels; (iii) ideal and incompressible gases; and (iv) electrochemical reactions confined to the electrode/electrolyte interfaces. The mathematical
The modeling study is performed to identify the important features of the tubular SIS-SOFC with SGC-IC films and to
Acknowledgment This work was supported by the Materials Technology Development Program (Development of Highly Conductive Nanocomposite Materials for Interconnects, No. 10037312) and the New & Renewable Energy Development Program (No. 20113020030010) funded by the Ministry of Knowledge Economy (MKE, Korea). Appendix
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B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
model is based on the conservation of momentum, mass and electrical charge, coupled with appropriate constitutive laws [15]. The gas flow in the channels is modeled by solving the Naviere Stokes equation (conservation of momentum) (A.1) and the continuity equation (A.2):
hact;a ¼ fele fion ;
(A.13)
hact;c ¼ fele fion Vrev ;
(A.14)
ru$Vu ¼ VP þ mV2 u;
(A.1)
where,
Vu ¼ 0:
(A.2)
Vrev
Convective and diffusive mass transport is considered in the channels:
and,
u$Vci þ V Dij $Vci ¼ 0:
Vo ¼ 1:253 2:4516 104 T:
(A.3)
The binary diffusion coefficient Dij of a gas mixture in Eq. (A.3) is written as follows:
Dij ¼ C
1 1 þ Mi Mj
!1 2
3
T2 ; ~ij UD Ps
(A.4)
where,
s~ij ¼ s~i þ s~j 2:
(A.5)
Mass transport in the porous electrodes includes free molecular and Knudsen diffusion modes, and is treated by the Dusty-gas model. The effective diffusion coefficient is given by,
Deff i;j ¼
1 1 þ s Dij Di;K 3
!1 ;
(A.6)
where the Knudsen diffusion coefficient is defined as,
Di;K ¼
1 1 8RT 2 : dp pMi 3
(A.7)
Charge transport is modeled by Ohm’s law (Eq. (A.8)) and the charge conservation equation (Eq. (A.9)):
j ¼ sVf;
(A.8)
Vjele ¼ 0:
(A.9)
The current density distribution at the electrode/electrolyte interface is described by the ButlereVolmer equation:
anele F ð1 aÞnele F hact ; j$n ¼ jo exp hact exp RT RT (A.10) where the exchange current density is represented by,
jo;a
¼ ga
PH2 Pref
!
! PH2 O Eact;a ; exp Pref RT
(A.11)
0 1 1 2 RT @PH2 PO2 A ¼ Vo þ ln ; PH2 O 2F
[1] N.Q. Minh, T. Takahashi, Science and Technology of Ceramic Fuel Cells, Elsevier, Amsterdam, 1995. [2] S.C. Singhal, K. Kendall, High Temperature Solid Oxide Fuel Cells e Fundamentals, Design and Applications, Elsevier, Amsterdam, 2003. [3] A.O. Isenberg, Solid State Ionics 3e4 (1981) 431. [4] P. Costamagna, A. Selimovic, M. Del Borghi, G. Agnew, Chem. Eng. J. 102 (2004) 61. [5] K. Tomida, N. Hisatome, T. Kabata, H. Tsukuda, A. Yamashita, Y. Yamazaki, Electrochemistry 77 (2009) 379. [6] W.Z. Zhu, S.C. Deevi, Mater. Sci. Eng. A 348 (2003) 227. [7] N. Sakai, H. Yokokawa, T. Horita, K. Yamaji, Int. J. Appl. Ceram. Technol. 1 (2004) 23. [8] B.-K. Park, J.-W. Lee, S.-B. Lee, T.-H. Lim, S.-J. Park, R.-H. Song, W.B. Im, D.R. Shin, Int. J. Hydrogen Energy 37 (2012) 4319. [9] U.-J. Yun, J.-W. Lee, S.-B. Lee, T.-H. Lim, S.-J. Park, R.-H. Song, D.-R. Shin, Fuel Cells 12 (2012) 1099. [10] X. Deng, J. Duquette, A. Petric, Int. J. Appl. Ceram. Technol. 4 (2007) 145. [11] S.-H. Pi, S.-B. Lee, R.-H. Song, J.-W. Lee, T.-H. Lim, S.-J. Park, D.-R. Shin, C.O. Park, Fuel Cells 13 (2013) 392. [12] M.R. Pillai, D. Gostovic, I. Kim, S.A. Barnett, J. Power Sources 163 (2007) 960. [13] H.-J. Son, R.-H. Song, T.-H. Lim, S.-B. Lee, S.-H. Kim, D.-R. Shin, J. Power Sources 195 (2010) 1779. [14] D.H. Jeon, Electrochim. Acta 54 (2009) 2727. [15] D. Cui, M. Cheng, J. Power Sources 195 (2010) 1435.
Nomenclature c: concentration d: diameter D: diffusion coefficient Eact: activation energy F: Faraday constant j: current density jo: exchange current density l: length M: molecular weight n: number of species n: unit vector normal to the boundary P: pressure r: radius R: gas constant Rcell: internal cell resistance t: thickness T: absolute temperature u: velocity vector V: voltage W: power density Greek symbols
jo;c ¼ gc
!0:25
3:
Eact;c exp : RT
(A.12)
The activation overpotentials for the anode and the cathode are written as follows:
(A.16)
References
a: transfer coefficient
PO2 Pref
(A.15)
porosity
f: potential g: pre-exponential coefficient h: polarization m: viscosity r: density s: conductivity s~: collision diameter
B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
s: tortuosity UD: collision integral Subscripts or superscripts a: anode act: activation ag: anode gap c: cathode cg: cathode gap e: electrolyte eff: effective
ele: electronic i: species ic: interconnect ion: ionic j: species K: Knudsen diffusion max: maximum p: pore ref: reference rev: reversible s: support
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Contents lists available at ScienceDirect
Current Applied Physics journal homepage: www.elsevier.com/locate/cap
A tubular segmented-in-series solid oxide fuel cell with metallic interconnect films: A performance study through mathematical simulations Bu-Won Son a, b,1, Seok-Joo Park a, Seung-Bok Lee a, Tak-Hyoung Lim a, Rak-Hyun Song a, Jong-Won Lee a, b, * a b
New and Renewable Energy Research Division, Korea Institute of Energy Research, 152 Gajeong-ro, Yuseong-gu, Daejeon 305-343, Republic of Korea Department of Advanced Energy Technology, University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 305-350, Republic of Korea
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 January 2013 Received in revised form 11 July 2013 Accepted 31 July 2013 Available online 9 August 2013
In a segmented-in-series solid oxide fuel cell (SIS-SOFC), an interconnect (IC) provides electrical contact and sealing between the anode of one cell and the cathode of the next. A metallic silver-glass composite (SGC) is considered a promising alternative to ceramic IC materials in SIS-SOFCs. In this work, a simulation study is performed on a tubular SIS-SOFC to assess the effectiveness of the SGC-IC design and to predict the SOFC performance characteristics for various IC geometries and conductivities. The developed model provides detailed information on cell behavior, such as the internal resistance, the potential/ current distribution, and the local gas species concentration. The results demonstrate that the SGC material greatly reduces a potential drop across the IC film. Thus, it provides the following substantial advantages over conventional ceramic IC materials: (i) increased power density and (ii) a larger degree of flexibility in the cell design. Moreover, the validation test, i.e., comparison of the simulated results with the experimental data, indicates that the model could serve as a valuable tool for design optimization to achieve the required SOFC performance. Ó 2013 Elsevier B.V. All rights reserved.
Keywords: Solid oxide fuel cell Segmented-in-series Mathematical model Interconnect Silver-glass composite
1. Introduction Solid oxide fuel cells (SOFCs) have received increasing attention in recent years because they have higher thermodynamic efficiency for energy conversion than low-temperature fuel cells (e.g., polymer electrolyte membrane fuel cells). In addition, SOFCs can be operated with a wide range of fuels, including natural gas, liquid hydrocarbons, coals, and bio-derived fuels [1,2]. A segmented-inseries (SIS) SOFC is an advanced design consisting of segmented unit cells connected in both electrical and gas flow series [3]. The unit cells are usually fabricated as a thin banded structure on a porous inert support tube, either tubular or flat-tubular, and they are electrically connected by interconnect (IC) layers [4,5]. The most attractive feature of the SIS-SOFC design is that the series
* Corresponding author. New and Renewable Energy Research Division, Korea Institute of Energy Research, 152 Gajeong-ro, Yuseong-gu, Daejeon, 305-343, Republic of Korea. Tel.: þ82 42 860 3025; fax: þ82 42 860 3297. E-mail address: [email protected] (J.-W. Lee). 1 Present address: Corporate R&D, Future Technology Center, LG Chem, 104-1, Moonji-dong, Yuseong-gu, Daejeon 305-738, Republic of Korea. 1567-1739/$ e see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cap.2013.07.025
connection of multiple cells on a single support tube builds up the total output voltage, leading to improved stack efficiency. In SIS-SOFCs, the IC layer electrically connects segmented unit cells and separates fuel (on the anode side) from oxidant (on the cathode side). The IC layer should be not only impermeable to gases, but also conductive and stable in a dual atmosphere (a reducing atmosphere on the anode side and an oxidizing atmosphere on the cathode side). Most IC films in SIS-SOFCs to date have been made of ceramic materials, such as doped LaCrO3 or SrTiO3 perovskites. Ca or Sr-doped LaCrO3 perovskites show acceptable conductivity and stability in the fuel cell environment [6,7]; however, it is quite difficult to fabricate a dense IC film on a porous support due to the poor sinterability of chromites. La-doped SrTiO3 perovskites have been adopted as an alternative IC material to doped chromites, as in the tubular SIS-SOFCs developed by Mitsubishi Heavy Industries Ltd. [5]. However, doped titanates are known to exhibit quite low electrical conductivity in a dual atmosphere, resulting in an increased internal resistance [8]. To address these problems, we recently developed a new tubular SIS-SOFC design, in which segmented unit cells are electrically connected by metallic IC films [9]. The IC films are fabricated
B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
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Fig. 1. Schematic diagrams of tubular SIS-SOFCs with (a) ceramic IC films and (b) SGC-IC films used for the simulation analysis. The drawings are not to scale.
using a silver-glass composite (SGC) material that is known as a conductive sealant in SOFCs [10,11]. Our previous study [11] showed that the electrical conductivity of the SGC material varies between 8.3 103 and 3.5 104 S m1 at 1023 K, depending on the glass content in the composite, while providing gas-tight sealing. Furthermore, the feasibility of the SGC-IC film was demonstrated on a tubular SIS-SOFC having 10 segmented cells. A power density as high as 3.3 103 W m2 (based on the active area) was achieved at 1023 K, and there was no performance degradation over 1100 h of continuous operation at 923 K [9]. No comprehensive optimization study was carried out, but it is expected that further optimizations of the cell and IC designs will lead to improved cell performance. The information provided by the previous experimental work [9] was limited to the fabrication procedure and the overall performance of the SIS-SOFC with metallic SGC-IC films. A deeper understanding of physical and electrochemical phenomena relevant to cell performance is still lacking. Modeling studies would be helpful in understanding the significance of the new IC design features and in determining the optimal IC geometries and the performance characteristics. With the objective of providing more information on the new IC design and the cell performance, this paper reports a modeling study of the tubular SIS-SOFC with metallic SGC-IC films. Firstly, a mathematical model for a tubular SIS-SOFC was developed that considers the electrochemical reaction, charge transport, and gas-phase mass transport. Secondly, the cell behavior was simulated for conventional ceramic IC films and metallic SGC-IC films to assess the effectiveness of the SGC-IC. Thirdly, the simulation was performed to determine the effects of IC geometry and conductivity on the performance, thereby making
predictions concerning optimal geometries and materials properties. Finally, the simulation data were compared with the experimental results for the SIS-SOFC fabricated using SGC-IC films. The distributions of electrical potential and gas species concentration in the SIS-SOFC, which cannot be determined by the experiments,
Table 1 Key parameters used in the simulation model. Parameter Cell geometries Unit cell length, lcell (mm) Anode gap length, lag (mm) Cathode gap length, lcg (mm) Support thickness, ts (mm) Anode thickness, ta (mm) Electrolyte thickness, te (mm) Cathode (1) thickness, tc1 (mm) Cathode (2) thickness, tc2 (mm) Outer radius of the support, rs (mm) Materials and electrochemical properties Anode conductivity, sa (S m1) Electrolyte conductivity, se (S m1) Cathode (1) conductivity, sc1 (S m1) Cathode (2) conductivity, sc2 (S m1) Transfer coefficient, a Anodic pre-exponential coefficient, ga (A m2) Cathodic pre-exponential coefficient, gc (A m2) Anodic activation energy, Eact,a (kJ mol1) Cathodic activation energy, Eact,c (kJ mol1) Tortuosity, s Porosity, 3 Average pore diameter, dp (mm)
Value 8 0.25 0.25 1.8 20 20 15 15 5 1 105 2 5 103 1.5 104 0.5 5.5 1010 7.0 109 120 120 2.5 0.45 1
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were quantitatively conditions.
B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
estimated
under
real
SOFC
operating
2. Cell geometries and simulation model Fig. 1(a) and (b) shows schematic diagrams of the tubular SISSOFCs having the ceramic and SGC-IC films, respectively, used for the simulation analysis. Each unit cell is divided into two regions: (i) the active region where the anode, the electrolyte, and the cathode overlap, and (ii) the inactive region consisting of the anode gap, the interconnect, and the cathode gap. The cathode is assumed to be composed of two layers, designated as cathode (1) and cathode (2). Cathode (1) represents a cathode functional layer in which the electrochemical oxygen reaction takes place, and cathode (2) corresponds to a current collecting layer to facilitate the lateral current flow in the plane of the cathode. Such a bi-layered cathode design has typically been employed in SIS-SOFCs to reduce ohmic losses in the cathode [9,12]. As shown in Fig. 1, the inactive part in the segmented unit cell varies in configuration, depending on whether ceramic material or SGC is used for the IC film. This is mainly due to a restriction on the fabrication process imposed by the sintering temperature of the IC material used: (i) The sintering temperatures of ceramic IC materials, such as doped LaCrO3 and SrTiO3 (1573e1873 K) are generally higher than those of cathode materials (1273e1473 K). For this reason, the ceramic IC film is co-sintered with the support, anode, and electrolyte, followed by the cathode coating and sintering. The ceramic IC film is, therefore, sandwiched between the anode and the cathode, and its entire surface is covered by the cathode, as presented in Fig. 1(a).
(ii) The SGC-IC film, on the other hand, should be coated and sintered after cathode sintering, because the melting point of silver (1235 K) is lower than the sintering temperatures of cathode materials. The SGC-IC film is fabricated so that it is in contact with the cross-sectional area of the cathode, as illustrated in Fig. 1(b). Unless otherwise noted, the anode gap length (lag) and the cathode gap length (lcg) are fixed as 0.25 mm as well, which could be practically achieved using a screen-printing technique. The outer radius (rs) and thickness (ts) of the porous support tube are 5 mm and 1.8 mm, respectively. The fuel is supplied to a channel on the left-hand side of the support tube. It diffuses through the porous support toward the anode, while the oxidant flows over the cathode. During fuel cell operations, the current flows along the anode of the cell, across the electrolyte, and along the cathode. The current then goes across the IC layer to the anode of the next cell. Given the radial symmetry of the tubular design, a 2dimensional model was used. The partial differential equations were described in cylindrical coordinates and solved in 2 dimensions (radial and axial) under the assumption that there are no significant circumferential variations in the boundary conditions. The mathematical model involves (i) the electrochemical reaction at the anode/electrolyte and electrolyte/cathode interfaces, (ii) electronic conduction in the electrodes, the interconnects, and the current collectors, (iii) ionic conduction in the electrolyte, and (iv) gas-phase diffusion in the electrodes, and (v) gas-phase diffusione convection in the channels. The governing and constitutive equations used in the model are summarized in the Appendix. A finite element computational package, COMSOL MULTIPHYSICSÒ, was used to solve the non-linear system of equations. The simulations
Fig. 2. Color maps of the electrical potential distribution in the inactive region of the anode gap and IC film simulated for (a) the ceramic IC film and (b) the SGCeIC film. The current path is also indicated as a streamline. The drawings are not to scale. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
were run at 1023 K using H2 gas with 3 vol.% H2O at the anode and air at the cathode. 3. Fabrication of a tubular SIS-SOFC with SGC-IC films The detailed fabrication process of a tubular SIS-SOFC with SGCIC films can be found elsewhere [9]. Briefly, a tubular support was produced by extrusion of the powder mixture: 3 mol% Y2O3-stabilized ZrO2 (3YSZ), activated carbon (YP-50F), and an organic binder (YB-131D). The extruded support was pre-sintered at 1373 K. An anode layer of NiOe8YSZ was coated onto the presintered support by a dip-coating process and then sintered at 1273 K. An 8YSZ electrolyte layer was formed by a vacuum slurry coating process [13] followed by sintering at 1673 K. A bi-layered cathode composed of (La0.85Sr0.15)0.9MnO3e8YSZ composite and La0.6Sr0.4Co0.2Fe0.8O3 was coated onto the co-sintered tube by a dipcoating process, and then it was sintered at 1423 K. Finally, the IC film was fabricated with the SGC material containing 90 wt.% Ag and 10 wt.% glass. The SGC paste was screen-printed onto the surface of the anode support without the coated electrolyte layer and then sintered at 973 K. 4. Results and discussion To understand how the IC’s design and conductivity affect the cell behavior, firstly, the simulations were performed on the SISSOFCs in which 5 segmented cells were connected in series by
Fig. 3. (a) Electrical potential profiles of the electrodes in the active regions and (b) the values of the current density norm within the electrolyte, calculated for the ceramic and SGC-IC films.
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either the ceramic or SGC-IC films. The key parameters, such as cell geometries and materials/electrochemical parameters, are listed in Table 1. Fig. 2(a) and (b) illustrates the electrical potential distribution (color map) along with the current path (streamline) in the inactive region of the anode gap and IC film, calculated for the ceramic and SGC-IC films, respectively. The values of lactive and lcell were taken as 7.25 and 8.00 mm, respectively, i.e., (lactive/lcell) ¼ 0.9375. As an example, the conductivities (sic) of the ceramic and SGC-IC materials were assumed to be 10 and 3.5 104 S m1, respectively. The simulations were carried out at the unit cell voltage, Vcell ¼ 0.5 V (total out voltage ¼ 2.5 V). The color map of the electrical potential distribution shows that the potential has the lowest value on the anode side where the current enters the unit cell, and it becomes higher on the cathode side where the current exits the cell. A potential difference as large as 0.222 V is observed across the ceramic IC film (Fig. 2(a)). However, the potential distribution across the SGC-IC film becomes more uniform (potential drop ¼ 0.036 V), as shown in Fig. 2(b), even though the thickness of the SGC-IC film (50 mm) is larger than that of the ceramic IC film (20 mm). As seen in Fig. 2, the current goes straight up through the ceramic IC film, whereas the current in the SGC-IC film runs mostly parallel to the xaxis. Fig. 3(a) compares the electrical potential profiles of the electrodes in the active region for the ceramic and SGC-IC films. The (lactive/lcell) value was fixed as 0.9375. The electric potentials of both
Fig. 4. Plots of (a) the maximum power density (Wmax) and (b) the internal cell resistance (Rcell) against the lactive-to-lcell ratio simulated for the SIS-SOFC with the ceramic IC films.
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the cathode and anode decrease with increasing x from 0 to lactive. The potential difference between the cathode and anode at x ¼ lactive is 0.63 V for the SGC-IC film as compared to 0.79 V for the ceramic IC film. This is due to a smaller potential drop across the SGC-IC film as shown in Fig. 2. To gain information about the distribution of the current crossing the electrolyte, the values of the current density norm are plotted against x in Fig. 3(b). The current density flowing across the electrolyte falls with increasing x, and goes through a shallow minimum at x ¼ ca. 2 mm, where the electrode potential difference deviates the least from the open circuit voltage. Then the current density increases with further increasing x, which indicates that most of the current travels through the anode and crosses the electrolyte at larger x to avoid a long current path across the more resistive cathode. The current density for the SGC-IC film remains higher over the whole electrolyte region compared to that for the ceramic IC film. This is ascribed to the fact that, as discussed in Fig. 2, the potential variation across the SGC-IC film is smaller than that across the ceramic IC film. Fig. 4(a) presents the maximum power density (Wmax) of the SIS-SOFC with the ceramic IC films as a function of (lactive/lcell). The unit cell length (lcell) was assumed to be 8 mm. Note that the calculated power was divided by the total area including both the active and inactive regions. The simulations were conducted using typical sic values of doped chromites or titanates (sic ¼ 1e 100 S m1). For sic ¼ 1 S m1, Wmax reaches a maximum value at
Fig. 6. (a) Comparison of the polarization curves of the SIS-SOFC (5 segmented cells) with the SGCeIC films simulated theoretically and measured experimentally, and (b) the color map of the electrical potential distribution simulated under experimental conditions. The drawings (b) are not to scale. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(lactive/lcell) ¼ ca. 0.53 and then decreases with further increasing (lactive/lcell). As sic increases, the (lactive/lcell) value at which Wmax shows a maximum shifts toward a higher value. To acquire a proper understanding about the cell behavior in Fig. 4(a), the internal cell resistance (Rcell) was computed as a function of (lactive/lcell), and the results are presented in Fig. 4(b). Here, Rcell is considered to be the sum of the resistances of the active and inactive parts. As shown in Fig. 4(b), Rcell falls rapidly with increasing (lactive/lcell), which is due to decreased resistance of the active region. When (lactive/lcell) exceeds ca. 0.78, a significant increase in the cell resistance with (lactive/lcell) is observed, particularly, for sic ¼ 1 S m1. This might result from (i) an increased IC resistance (a reduced cross-sectional area of the IC film) and (ii) an increased electrode resistance (an increased electrode length). The occurrence of local maxima on the Wmax vs. (lactive/lcell) curves in Fig. 4(a), therefore, could be understood qualitatively in terms of a trade-off between an increase in the active area and an increase in the resistances of the electrodes and IC film. Fig. 5(a) and (b) shows the plots of Wmax and Rcell vs. (lactive/lcell), respectively, computed with various sic values for the SGC-IC films. Note that Wmax increases continuously with increasing (lactive/lcell) without showing any local maxima, regardless of the sic values used for the simulation. Moreover, Rcell remains almost constant at (lactive/lcell) > ca. 0.16. As a result, the simulation data in Figs. 4 and 5 clearly demonstrate that the SGC-IC film with high electrical conductivity offers two major benefits over the ceramic IC film: (i) increased power density and (ii) a larger degree of flexibility in the cell design (no performance loss for high (lactive/lcell) values).
Table 2 Cell geometries modified in the model for comparison with the experimental data.
Fig. 5. Plots of (a) the maximum power density (Wmax) and (b) the internal cell resistance (Rcell) against the lactive-to-lcell ratio simulated for the SIS-SOFC with the SGCeIC films.
Parameter
Value
Unit cell length, lcell (mm) Active cell length, lactive (mm) Anode gap length, lag (mm) Cathode gap length, lcg (mm)
16 6 1 1
B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
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Fig. 7. Distributions of (a) H2, (b) O2 and (c) H2O species within the fabricated SIS-SOFC with 5 segmented cells under experimental conditions. The drawings are not to scale. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The polarization curve (cell voltage vs. current density) simulated for the SIS-SOFC with the SGC-IC films is compared with the experimental one in Fig. 6(a). The fabricated SIS-SOFC consists of 5 segmented unit cells. Note that the power was divided by the total unit cell area including both the active and inactive regions. The experimental measurements were conducted at 1023 K using the constant flow rate mode with the H2 and air flow rates of 300 and 500 mL min1, respectively. The cell geometries used for the simulation were modified to match those of the fabricated cell, as summarized in Table 2. As shown in Fig. 6, the calculated polarization curve is in good accord with the experimental data, which confirms that the equations and parameters used in the SIS-SOFC model are reasonable. The electrical potential distribution of the fabricated SIS-SOFC was calculated under experimental conditions and is given in Fig. 6(b), in which the potential deceases from the 5th unit cell (right) to the 1st (left). To illustrate the gas species concentrations within the fabricated SIS-SOFC under practical operating conditions, 3-dimensional representations of H2, O2 and H2O concentrations are shown in Fig. 7(a)e(c), respectively. As expected, the highest concentration values are located at the cell inlet, and progressively decreasing H2 and O2 concentrations are observed moving toward the cell outlet. In the channels, the concentration variations of gas species are small due to relatively high mass flow rates, while a steep gradient is observed in the electrode due to consumption (or generation) of gas species and low permeability caused by low effective diffusivity [14].
determine the performance characteristics for various IC geometries and conductivities. The developed model is used to calculate the polarization behavior (potential vs. current curves) as well as the internal resistance, the potential/current distribution, and the local gas species concentration under SOFC operating conditions. The results simulated for ceramic and SGC-IC films indicate that when the SGC material is used for IC films, the potential drop across the IC film is greatly reduced; therefore, the SGC-IC offers advantages over conventional ceramic IC films, namely, increased power density and a larger degree of flexibility in the cell design. The power density is also found to increase with decreasing SGCIC length and with increasing conductivity. The validation test shows that the model precisely predicts the cell behavior; thus, it can be used as a tool for design optimization to improve SOFC performance.
5. Conclusions
The physical model used in this study assumes the following: (i) steady-state conditions; (ii) laminar gas flow in the channels; (iii) ideal and incompressible gases; and (iv) electrochemical reactions confined to the electrode/electrolyte interfaces. The mathematical
The modeling study is performed to identify the important features of the tubular SIS-SOFC with SGC-IC films and to
Acknowledgment This work was supported by the Materials Technology Development Program (Development of Highly Conductive Nanocomposite Materials for Interconnects, No. 10037312) and the New & Renewable Energy Development Program (No. 20113020030010) funded by the Ministry of Knowledge Economy (MKE, Korea). Appendix
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B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
model is based on the conservation of momentum, mass and electrical charge, coupled with appropriate constitutive laws [15]. The gas flow in the channels is modeled by solving the Naviere Stokes equation (conservation of momentum) (A.1) and the continuity equation (A.2):
hact;a ¼ fele fion ;
(A.13)
hact;c ¼ fele fion Vrev ;
(A.14)
ru$Vu ¼ VP þ mV2 u;
(A.1)
where,
Vu ¼ 0:
(A.2)
Vrev
Convective and diffusive mass transport is considered in the channels:
and,
u$Vci þ V Dij $Vci ¼ 0:
Vo ¼ 1:253 2:4516 104 T:
(A.3)
The binary diffusion coefficient Dij of a gas mixture in Eq. (A.3) is written as follows:
Dij ¼ C
1 1 þ Mi Mj
!1 2
3
T2 ; ~ij UD Ps
(A.4)
where,
s~ij ¼ s~i þ s~j 2:
(A.5)
Mass transport in the porous electrodes includes free molecular and Knudsen diffusion modes, and is treated by the Dusty-gas model. The effective diffusion coefficient is given by,
Deff i;j ¼
1 1 þ s Dij Di;K 3
!1 ;
(A.6)
where the Knudsen diffusion coefficient is defined as,
Di;K ¼
1 1 8RT 2 : dp pMi 3
(A.7)
Charge transport is modeled by Ohm’s law (Eq. (A.8)) and the charge conservation equation (Eq. (A.9)):
j ¼ sVf;
(A.8)
Vjele ¼ 0:
(A.9)
The current density distribution at the electrode/electrolyte interface is described by the ButlereVolmer equation:
anele F ð1 aÞnele F hact ; j$n ¼ jo exp hact exp RT RT (A.10) where the exchange current density is represented by,
jo;a
¼ ga
PH2 Pref
!
! PH2 O Eact;a ; exp Pref RT
(A.11)
0 1 1 2 RT @PH2 PO2 A ¼ Vo þ ln ; PH2 O 2F
[1] N.Q. Minh, T. Takahashi, Science and Technology of Ceramic Fuel Cells, Elsevier, Amsterdam, 1995. [2] S.C. Singhal, K. Kendall, High Temperature Solid Oxide Fuel Cells e Fundamentals, Design and Applications, Elsevier, Amsterdam, 2003. [3] A.O. Isenberg, Solid State Ionics 3e4 (1981) 431. [4] P. Costamagna, A. Selimovic, M. Del Borghi, G. Agnew, Chem. Eng. J. 102 (2004) 61. [5] K. Tomida, N. Hisatome, T. Kabata, H. Tsukuda, A. Yamashita, Y. Yamazaki, Electrochemistry 77 (2009) 379. [6] W.Z. Zhu, S.C. Deevi, Mater. Sci. Eng. A 348 (2003) 227. [7] N. Sakai, H. Yokokawa, T. Horita, K. Yamaji, Int. J. Appl. Ceram. Technol. 1 (2004) 23. [8] B.-K. Park, J.-W. Lee, S.-B. Lee, T.-H. Lim, S.-J. Park, R.-H. Song, W.B. Im, D.R. Shin, Int. J. Hydrogen Energy 37 (2012) 4319. [9] U.-J. Yun, J.-W. Lee, S.-B. Lee, T.-H. Lim, S.-J. Park, R.-H. Song, D.-R. Shin, Fuel Cells 12 (2012) 1099. [10] X. Deng, J. Duquette, A. Petric, Int. J. Appl. Ceram. Technol. 4 (2007) 145. [11] S.-H. Pi, S.-B. Lee, R.-H. Song, J.-W. Lee, T.-H. Lim, S.-J. Park, D.-R. Shin, C.O. Park, Fuel Cells 13 (2013) 392. [12] M.R. Pillai, D. Gostovic, I. Kim, S.A. Barnett, J. Power Sources 163 (2007) 960. [13] H.-J. Son, R.-H. Song, T.-H. Lim, S.-B. Lee, S.-H. Kim, D.-R. Shin, J. Power Sources 195 (2010) 1779. [14] D.H. Jeon, Electrochim. Acta 54 (2009) 2727. [15] D. Cui, M. Cheng, J. Power Sources 195 (2010) 1435.
Nomenclature c: concentration d: diameter D: diffusion coefficient Eact: activation energy F: Faraday constant j: current density jo: exchange current density l: length M: molecular weight n: number of species n: unit vector normal to the boundary P: pressure r: radius R: gas constant Rcell: internal cell resistance t: thickness T: absolute temperature u: velocity vector V: voltage W: power density Greek symbols
jo;c ¼ gc
!0:25
3:
Eact;c exp : RT
(A.12)
The activation overpotentials for the anode and the cathode are written as follows:
(A.16)
References
a: transfer coefficient
PO2 Pref
(A.15)
porosity
f: potential g: pre-exponential coefficient h: polarization m: viscosity r: density s: conductivity s~: collision diameter
B.-W. Son et al. / Current Applied Physics 13 (2013) 1906e1913
s: tortuosity UD: collision integral Subscripts or superscripts a: anode act: activation ag: anode gap c: cathode cg: cathode gap e: electrolyte eff: effective
ele: electronic i: species ic: interconnect ion: ionic j: species K: Knudsen diffusion max: maximum p: pore ref: reference rev: reversible s: support
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